Bootstrap tests: how many bootstraps?

In practice, bootstrap tests must use a finite number of bootstrap samples. This means that the outcome of the test will depend on the sequence of random numbers used to generate the bootstrap samples, and it necessarily results in some loss of power. We examine the extent of this power loss and propose a simple pretest procedure for choosing the number of bootstrap samples so as to minimize experimental randomness. Simulation experiments suggest that this procedure will work very well in practice.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Dependence Estimation Over a Finite Bivariate Failure Time Region

This articleconcerns nonparametric estimation of association between bivariatefailure times. In the presence of independent right censoring,the support for failure time variates may be restricted and measuresof dependence over a finite failure time region may be of particularinterest. To this end, the reciprocal cross ratio function, weightedby the bivariate failure time density, is proposed as a summarymeasure of dependence over a failure time region. This `relativerisk' estimator is shown to be consistent and asymptoticallynormally distributed, with consistent bootstrap variance estimator.A finite-region version of Kendall's tau, which is suitable forcensored failure time data, is also proposed, and correspondingasymptotic distribution theory is noted. The accuracy of theseasymptotic approximations is studied in simulations and an illustrationis provided.     

Designing and integrating composite networks for monitoring multivariate gaussian pollution fields

Networks of ambient monitoring stations are used to monitor environmental pollution fields such as those for acid rain and air pollution. Such stations provide regular measurements of pollutant concentrations. The networks are established for a variety of purposes at various times so often several stations measuring different subsets of pollutant concentrations can be found in compact geographical regions. The problem of statistically combining these disparate information sources into a single 'network' then arises. Capitalizing on the efficiencies so achieved can then lead to the secondary problem of extending this network. The subject of this paper is a set of 31 air pollution monitoring stations in southern Ontario. Each of these regularly measures a particular subset of ionic sulphate, sulphite, nitrite and ozone. However, this subset varies from station to station. For example only two stations measure all four. Some measure just one. We describe a Bayesian framework for integrating the measurements of these stations to yield a spatial predictive distribution for unmonitored sites and unmeasured concentrations at existing stations. Furthermore we show how this network can be extended by using an entropy maximization criterion. The methods assume that the multivariate response field being measured has a joint Gaussian distribution conditional on its mean and covariance function. A conjugate prior is used for these parameters, some of its hyperparameters being fitted empirically.     

Estimating the propagation rate of a viral infection of potato plants via mixtures of regressions

A problem arising from the study of the spread of a viral infection among potato plants by aphids appears to involve a mixture of two linear regressions on a single predictor variable. The plant scientists studying the problem were particularly interested in obtaining a 95% confidence upper bound for the infection rate. We discuss briefly the procedure for fitting mixtures of regression models by means of maximum likelihood, effected via the EM algorithm. We give general expressions for the implementation of the M-step and then address the issue of conducting statistical inference in this context. A technique due to T. A. Louis may be used to estimate the covariance matrix of the parameter estimates by calculating the observed Fisher information matrix. We develop general expressions for the entries of this information matrix. Having the complete covariance matrix permits the calculation of confidence and prediction bands for the fitted model. We also investigate the testing of hypotheses concerning the number of components in the mixture via parametric and 'semiparametric' bootstrapping. Finally, we develop a method of producing diagnostic plots of the residuals from a mixture of linear regressions.     

Two ways of modelling overdispersion in non-normal data

For non-normal data assumed to have distributions, such as the Poisson distribution, which have an a priori dispersion parameter, there are two ways of modelling overdispersion: by a quasi-likelihood approach or with a random-effect model. The two approaches yield different variance functions for the response, which may be distinguishable if adequate data are available. The epilepsy data of Thall and Vail and the fabric data of Bissell are used to exemplify the ideas.     

Asymptotic Analysis of the Sieve Estimator for a Class of Parabolic SPDEs

In this paper we consider the problem of estimating a coefficient of a strongly elliptic partial differential operator in stochastic parabolic equations. The coefficient is a bounded function of time. We compute the maximum likelihood estimate of the function on an approximating space (sieve) using a finite number of the spatial Fourier coefficients of the solution and establish conditions that guarantee consistency and asymptotic normality of the resulting estimate as the number of the coefficients increases. The equation is assumed diagonalizable in the sense that all the operators have a common system of eigenfunctions.     

Practical Maximum Pseudolikelihood for Spatial Point Patterns(with Discussion)

This paper describes a technique for computing approximate maximum pseudolikelihood estimates of the parameters of a spatial point process. The method is an extension of Berman & Turner's (1992) device for maximizing the likelihoods of inhomogeneous spatial Poisson processes. For a very wide class of spatial point process models the likelihood is intractable, while the pseudolikelihood is known explicitly, except for the computation of an integral over the sampling region. Approximation of this integral by a finite sum in a special way yields an approximate pseudolikelihood which is formally equivalent to the (weighted) likelihood of a loglinear model with Poisson responses. This can be maximized using standard statistical software for generalized linear or additive models, provided the conditional intensity of the process takes an 'exponential family' form. Using this approach a wide variety of spatial point process models of Gibbs type can be fitted rapidly, incorporating spatial trends, interaction between points, dependence on spatial covariates, and mark information.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.     

STEIN'S METHOD AND THE BERRY-ESSÉEN THEOREM

Stein's method is used to prove the Lindeberg-Feller theorem and a generalization of the Berry-Esséen theorem. The arguments involve only manipulation of probability inequalities, and form an attractive alternative to the less direct Fourier-analytic methods which are traditionally employed.